Truth Value



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Truth Value

In a manner similar to that for terms, we define the truth value for sentences in the language as a function that maps sentences into the truth values or .

The truth value of a logical constant is the truth value assigned by the corresponding interpretation.

An equation is true if and only if the terms in the equation refer to the same object in the universe of discourse.

An inequality is true if and only if the terms in the equation refer to distinct objects in the universe of discourse.

The truth value of a simple relational sentence without a terminating sequence variable is

if and only if the relation denoted by the relation constant in the sentence is true of the objects denoted by the arguments. Equivalently, viewing a relation as a set of tuples, we say that the truth value of a relational sentence is

if and only if the tuple of objects formed from the values of the arguments is a member of the set of tuples denoted by the relation constant.

If a relational sentence terminates in a sequence variable, the sentence is true if and only if the relation contains the tuple consisting of the values of the terms that precede the sequence variable together with the objects in the sequence denoted by the variable. Remember that the vertical bar

means that the objects in the sequence following the bar are appended to the sequence of elements before the bar.

The truth value of a negation is

if and only if the truth value of the negated sentence is

.

The truth value of a conjunction is

if and only if the truth value of every conjunct is

.

The truth value of a disjunction is

if and only if the truth value of at least one of the disjuncts is

.

If the truth value of every antecedent in an implication is

, then the the truth value of the implication as a whole is

if and only if the truth value of the consequent is

. If any of the antecedents is

, then the implication as a whole is

, regardless of the truth value of the consequent.

A reverse implication is just an implication with the consequent and antecedents reversed.

The truth value of an equivalence is

if and only if the embedded sentences have the same truth value.

Given an interpretation

and variable assignment

, the truth value of an existentially quantified sentence is

if and only if the truth value of the second argument is

for some version

of variable assignment

with respect to the variables in the first argument.

Given an interpretation

and variable assignment

, the truth value of a universally quantified sentence is

if and only if the truth value of the second argument of the sentence is

for every version

of

with respect to variables in the first argument.



next up previous
Next: Logical Entailment Up: Semantics Previous: Semantic Value



Vishal I. Sikka
Wed Dec 7 13:23:42 PST 1994